# Homework Help: Minimum and maximum uncertainty values in quantum harmonic oscillator

1. Jun 12, 2009

### cleggy

1. The problem statement, all variables and given/known data

I have to find the minimum and maximum values of the uncertainty of $$\Delta$$x and specify the times after t=0 when these uncertainties apply.

2. Relevant equations

The wave function is Ψ(x, 0) = (1/√2) (ψ1(x)+ iψ3(x))

and for all t is Ψ(x, t) = (1/√2) (ψ1(x)exp(-3iwt/2+ iψ3(x)exp(-7iwt/2)

3. The attempt at a solution

the expectation value <x> = 0 ( given in my text )

hence $$\Delta$$x= $$\sqrt{}<x^2>$$

using the sandwich integral

$$\int$$$$^{\infty}_{-\infty}$$ψ1$$\ast$$(x) x^2 ψ1(x) dx = $$\frac{3}{2}$$ a^2

$$\int$$$$^{\infty}_{-\infty}$$$$\psi$$3$$\ast$$(x) x^2 $$\psi$$3(x) dx = $$\frac{7}{2}$$a^2

$$\int$$$$^{\infty}_{-\infty}$$$$\psi$$3$$\ast$$(x) x^2 $$\psi$$1 (x) dx = $$\int$$$$^{\infty}_{-\infty}$$$$\psi$$$$\ast$$1(x) x^2 $$\psi$$3 dx = $$\sqrt{\frac{3}{2}}$$a^2

where a is the length parameter of the oscillator.

Where do I go from here?

Last edited: Jun 12, 2009
2. Jun 12, 2009

### Cyosis

You have calculated the probability density function correctly in a previous question you asked. Where has the time dependence disappeared? Secondly how does one find minimum/maximum values of a function generally?

3. Jun 12, 2009

### cleggy

I don't know where the time dependence has gone

I'm not sure how to calculate the minimum and maximum values of a function?

4. Jun 12, 2009

### cleggy

I've been going at this for so long now my mind is turning mushy

5. Jun 12, 2009

### Cyosis

If you're following a course on quantum mechanics I'm pretty sure you know how to find the minimum/maximum of a function.

Hint: derivative

6. Jun 12, 2009

### cleggy

so i have to do a first derivative test

7. Jun 12, 2009

### Cyosis

First you need to find the correct time dependent expression for <x^2>. After that you take the derivative with respect to the appropriate variable and solve for said variable.

8. Jun 12, 2009

### cleggy

I don't know how to find the time dependent expression for <x^2>

9. Jun 12, 2009

### Cyosis

In your other thread on this topic you calculated the probability density function for this time dependent wave function. The one with a sine in it and a t etc? You must remember.
This is the correct "sandwich" expression to use. What you've calculated now is the time independent expectation value of x^2.